Sequentially compact space

  • Thread starter Hymne
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Hello Physicsforums!
I have a problem with the difference between complete metric space and a sequentially compact metric space.
For the first one every Cauchy sequence converges inside the space, which is no problem.
But for the last one "every sequence has a convergent subsequence." (-Wiki) And it's here that I get lost.

How does this affect the constraints on the space?
Could someone please try to give me an intuitive explanation?

For [1,9] on the real axis we can take the sequence (1,2,3,4,5,6) as an example. How do we find a convergent subsequence in this one?
Have I missunderstood it all?
 

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  • #2
George Jones
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For [1,9] on the real axis we can take the sequence (1,2,3,4,5,6) as an example. How do we find a convergent subsequence in this one?
Have I missunderstood it all?
What is the definition of "sequence"?
 
  • #3
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What is the definition of "sequence"?
Hmm, I use this one http://en.wikipedia.org/wiki/Sequence .
With
In mathematics, a sequence is an ordered list of objects (or events). Like a set, it contains members (also called elements or terms), and the number of terms (possibly infinite) is called the length of the sequence. Unlike a set, order matters, and the exact same elements can appear multiple times at different positions in the sequence.
Maybe it´s here that I am confused. :uhh:

Should we only work with Cauchy sequences maybe?
 
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These definitions apply to infinite sequences. (1,2,3,4,5,6) is not an infinite sequence. It doesn't even mean anything for a finite sequence to converge!
 
  • #5
jav
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To the original question..

In a complete metric space (an) converges <-> (an) is cauchy

In a compact metric space, every sequence an contains a convergent subsequence (ank).

We should note that convergence -> cauchy in any metric space.

Then, in a compact metric space, every sequence an contains a cauchy subsequence (ank).

Regardless, the properties of these two types of spaces are completely different.

A simple example highlighting the difference between the two is a subset of R1. Consider, the interval (0,1).

By the Heine-Borel theorem, this space is not compact since it is not closed.

It is, however, a complete metric space since cauchy <-> convergent in R1.

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